The IAU has different classifications for objects in the solar system orbiting the sun, they can be according to their shape and if they clear their neighborhood of other material around its orbits , planets, dwarf planets or Small Solar System bodies. But for objects in the solar system orbiting a planet or a dwarf planet they are all satellites, no matter if they are rounded or if they have an irregular shape like an asteroid. Is there any known reason why this is so? Why is it needed to differenciate its shape when the object it's orbiting the sun, but it isnt needed when the object it's orbiting a planet? (or dwarf planet)?

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At one time (1980) things seemed clear enough: There were planets (Mercury-Pluto), Asteroids (mostly between Mars and Jupiter) and Satellites. No one care much about whether a body was spherical or not, since it was pretty clear that this was all there was.

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Then the Kuiper belt happened, and in particular it became clear that Eris was big enough to be rounded... Now why should not Eris be a planet? Could we have tens of rounded icy objects orbiting beyond Neptune? How many would we expect grade-schoolers to memorise?

The decision was taken to classify solar system bodies into:

Planets (round and not shareing their orbit with anything comparible in size)

Dwarf planets (round, but have not cleared their orbit)

Minor planets (not round)

Satellites (not orbiting the sun)

This is a matter of convenience for humans. It does not reflect any fundamental reality.

For convenience, there is no need to distinguish between round moons and non-round ones, so no distinction is needed (for humans)